The scope of this paper is to work on the relation between matrix-valued orthogonal polynomials and non-commutative integrable equations. This is a relatively new field, in which few examples have been worked out. In [1]}, it had been shown that matrix-valued orthogonal polynomials (on the real line) satisfy a non-commutative version of the Toda equations, and in [2]} one of the authors provided an analogue result for matrix-valued orthogonal polynomials on the unit circle, upon replacing Toda equations with the (multi-component) Ablowitz–Ladik hierarchy (a well known integrable discretisation of the non-linear Schrödinger one). In both cases, the main tool had been the theory of quasi-determinants (Schur complements), which were known to be related to non-commutative integrable systems since the pioneering work of Etingof, Gelfand and Retakh [3]}. Schur complements had been used also in the series of papers [4]}, [5]}, [6]} and in [7]}, where a non-commutative version of the Painlevé II equation was introduced, and studied later on in [8]} using Riemann–Hilbert techniques.
